## Game ideals.(English)Zbl 1173.03036

For $$\kappa$$ a regular uncountable cardinal and $$\lambda$$ a cardinal, $$\lambda>\kappa$$, let $$A\subseteq P_{\kappa}(\lambda)$$. The author considers the following two-player (I and II) game $$H_{\kappa, \lambda}(A)$$: there are $$\omega$$ moves and I goes first; I and II alternately choose members of $$P_{\kappa}(\lambda)$$. The result is a sequence $$\langle a_n : n<\omega\rangle$$. Player I wins if and only if $$\bigcup_{n<\omega} a_n\notin A$$. The set of all $$A$$ for which player I has a winning strategy in $$H_{\kappa, \lambda}(A)$$ is a normal ideal on $$P_{\kappa}(\lambda)$$, denoted by $$NG_{\kappa,\lambda}$$. The paper under review studies the properties of $$NG_{\kappa,\lambda}$$, using such tools as club-guessing, scales, Namba trees, and mutually stationary sets.
The author shows that $$P_{\kappa}(\lambda)$$ is the disjoint union of $$M(\kappa,\lambda)$$ sets not in $$NG_{\kappa,\lambda}$$, where $$M(\kappa,\lambda)$$ is $$\lambda^{\aleph_0}$$ if $$\text{cf}(\lambda)=\omega$$ or $$\text{cf}(\lambda)\geq\kappa$$, and $$\lambda^{+}\cdot \lambda^{\aleph_0}$$ otherwise. He also observes that if there is no inner model with $$\aleph_2$$ measurable cardinals, then $$M(\kappa,\lambda)$$ is the least size of any member of the dual filter $$NG_{\kappa,\lambda}^{*}$$, and so $$P_{\kappa}(\lambda)$$ cannot be decomposed into more than $$M(\kappa,\lambda)$$ members of $$NG_{\kappa,\lambda}^{+}$$. In the paper’s final section the author proves that if $$2^{<\kappa}\leq \mu^{\aleph_0}$$ for some regular cardinal $$\mu$$ with $$\kappa<\mu\leq\lambda$$, then $$\diamondsuit_{\kappa,\lambda}[NG_{\kappa,\lambda}]$$ holds.

### MSC:

 300000 Other combinatorial set theory 300 Partition relations

### Keywords:

$$P_\kappa(\lambda)$$; games; diamond principle
Full Text:

### References:

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